### Abstract

We consider birth-death processes taking values in N ≡ {0,1,... }, but allow the death rate in state 0 to be positive, so that escape from N is possible. Two such processes with transition functions { pij(t) } and { ̃pij(t) } are said to be similar if, for all i, j ∈ N, there are constants cij such that ̃pij(t) = cijpij(t) for all t ≥ 0. We determine conditions on the birth and death rates of a birth-death process for the process to be a member of a family of similar processes, and we identify the members of such a family. These issues are also resolved in the more general setting in which the two processes are called similar if there are constants cij and ν such that ̃pij(t) = cijeνtpij(t) for all t ≥ 0.

Original language | Undefined |
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Article number | 10.1239/jap/1014842840 |

Pages (from-to) | 835-849 |

Number of pages | 14 |

Journal | Journal of applied probability |

Volume | 2000 |

Issue number | 37 |

DOIs | |

Publication status | Published - 2000 |

### Keywords

- invariant vector
- EWI-12851
- transition function
- METIS-140634
- Transient behaviour
- IR-62348
- Chain sequence
- Absorption probability

## Cite this

Lenin, R. B., Parthasarathy, P. R., Scheinhardt, W. R. W., & van Doorn, E. A. (2000). Families of birth-death processes with similar time-dependent behaviour.

*Journal of applied probability*,*2000*(37), 835-849. [10.1239/jap/1014842840]. https://doi.org/10.1239/jap/1014842840